On the Enumeration of Inscribable Graphs

| We explore the question of counting, and estimating the number and the fraction of, in-scribable graphs. In particular we will concern ourselves with the number of inscribable and circum-scribable maximal planar graphs (synonym: simplicial polyhedra) on V vertices, or, dually, the number of circumscribable and inscribable trivalent (synonyms: 3-regular, simple) polyhedra. For small V we provide computer-generated tables. Asymptotically for large V we will prove bounds showing that these graphs are exponentially numerous, but, viewed as a fraction of all maximal planar graphs, they are exponentially rare. Many of our results are based on a lemma, the \strong 0-1 law for maximal planar graphs," of independent interest. This is part of a series of TMs exploring graph-theoretic consequences of the recent Rivin-Smith characterization of \inscribable graphs." (A graph is a set of \vertices," some pairs of which are joined by \edges." A graph is \inscribable" if it is the 1-skeleton of a convex polyhedron inscribed in a sphere, \circumscribable" if it is the 1-skeleton of a convex polyhedron each face of which is tangent to a common sphere.)